4D Discrete Lattice Proof of Concept: Emergent Relativistic Kinematics, Low-Energy Quantum Dynamics, and Regge-Based Gravitational Dynamics

We present a proof of concept in which space-time is modeled as a discrete four-dimensional lattice with a fundamental update step. The aim is not a complete unification scheme, but a controlled derivation of three linked emergent mechanisms. First, relativistic kinematics is obtained from structural transition constraints, so the velocity bound c and time dilation arise kinematically rather than as postulates. Second, the low-energy limit of a nearestneighbor tight-binding Hamiltonian recovers the Schrödinger equation with an effective inertial mass m ∝ (Jℓ2)−1. Third, smooth gradients of the link-tension field Txy define an effective Gordon-type metric in the weak-field regime, while the full Einstein dynamics is recovered through Regge calculus by extremizing a simplicial action built directly from Txy . The framework is falsifiable because the phenomenology is expressed as bounds on Lorentzinvariance-violation operators and anisotropy coefficients, avoiding ad hoc parameter fitting. In this way the model sits within the analog-gravity program while addressing the usual gap between kinematic emergence and dynamical gravitational field equations.